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## 数学代写|示性类代考Characteristic Classes代考|Homology theory and homotopy theory

Homology theory and homotopy theory. Among the methods of investigating geometrical properties of a given figure or space, we have homology theory and homotopy theory. The former was begun by Poincaré around 1900, and the latter was initiated by Hurewicz in the 1930’s. Briefly speaking, in homology theory we decompose a given figure into components like points, segments, triangles and in general $k$-dimensional simplices (the triangulation), and then we extract a topological invariant called the homology group out of the way they are connected to each other. There is another way of decomposing figures, namely by means of CW complexes which are more flexible than triangulations. Just to make sure, let us recall the definition of CW complexes. Let $D^n=\left{x \in \mathbb{R}^n ;|x| \leq 1\right}$ be the $n$-dimensional disk and let $S^{n-1}=\partial D^n=\left{x \in \mathbb{R}^n ;|x|=1\right}$ be its boundary, namely the $(n-1)$-dimensional sphere.

DEFINITION 1.1. Let $X$ be a topological space and let $f: S^{n-1} \rightarrow$ $X$ be a continuous map. We denote by
$$X \cup_f D^n$$
the space obtained from the disjoint union of $X$ and $D^n$ by identifying each point $x \in S^{n-1}$ with $f(x) \in X$. It is called the space obtained by attaching an $n$-cell $e^n=D^n \backslash S^{n-1}$ to $X$ by $f$ or simply the attaching space (see Figure 1.1). The map $f$ is called the attaching map.

## 数学代写|示性类代考Characteristic Classes代考|Postnikov decomposition

Postnikov decomposition. Given a topological space $X$, if we can construct a triangulation of it or a decomposition as a CW complex, then it is convenient for the study of homological structure. However, it is not so useful for homotopy theoretical study of the space. The Postnikov decomposition describes the homotopy type of $X$ in terms of Eilenberg-MacLane spaces as fundamental components. The simplest space whose homotopy groups are the same as those of $X$ would be
$$K\left(\pi_1(X), 1\right) \times K\left(\pi_2(X), 2\right) \times \cdots$$
which is the product of various Eilenberg-MacLane spaces. In general, $X$ is not the product but a certain twisted version of the above space. The way it is twisted is described by what is called the Postnikov invariants.

## 数学代写|示性类代考Characteristic Classes代考|Homology theory and homotopy theory

〈left 缺少或无法识别的分隔符

$$X \cup_f D^n$$

## 数学代写|示性类代考Characteristic Classes代考|Postnikov decomposition

Postnikov分解。给定一个拓扑空间 $X$ ，如果戔们可以将它构造一个三角剖分或者分解为一个CW貪形，那/对于同调结构的研究 就很方便了。然而，它对空间的同伦理论研究没有多大用处。Postnikov 分解渵术了的同伦类型 $X$ 在 Eilenberg-MacLane 空间 方面作为基本组成部分。同伦群相同的最简单空间 $X$ 将会
$$K\left(\pi_1(X), 1\right) \times K\left(\pi_2(X), 2\right) \times \cdots$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。